#6622762

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,23,29,37)(20,31,35,41)(21,25,32,26)(24,33,40,34)(28,42,44,48)(30,38,43,39)(36,45,47,46);(2,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,28)(21,30)(22,29)(24,36)(25,38)(26,39)(27,37)(31,42)(32,43)(33,45)(34,46)(35,44)(40,47)(41,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,32,33,22,35,21,34)(18,24,25,41,27,40,26,31)(19,28,43,45,29,44,30,46)(23,36,38,48,37,47,39,42);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,29)(23,37)(28,44)(30,43)(36,47)(38,39)(42,48)(45,46);(9,12,13,11)(10,16,14,15)(17,21,22,32)(18,25,27,26)(19,30,29,43)(20,33,35,34)(23,38,37,39)(24,31,40,41)(28,45,44,46)(36,42,47,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529958);